Monday, October 31, 2016

A pair of scary math-education anecdotes by Junaid Mubeen, for your consideration:

How Old is the Shepherd? When 8th-graders are asked a short question with absolutely no information about age whatsoever, 3-in-4 will report some numerical result anyway. Repeated in numerous experiments. Watch a video.

I Can't Believe It's Not Unproven. Mubeen's 12-year-old nephew comes home with a math problem that can't be solved; he is shown a proof of that fact, and agrees to all the steps and the conclusion. Nephew spends the rest of the evening trying to find an answer anyway.

I don't really agree with Mubeen's rather broad conclusions at the end of the first article. But we can all agree this is a terrifying outcome!

Monday, October 24, 2016

If you look at any list of elementary algebra topics, or any book's table of contents, etc., then you'll probably find that all of the subjects are referenced by name except for one single exceptional case, which is always expressed in symbolic form. For example, from the College Board's Accu-Placer Program Manual, here's a list of Content Areas for the Elementary Algebra test:

Do you see it? Or, here are some of the section headers in the Pearson testbank which accompanies the Martin-Gay Prealgebra & Introductory Algebra text:

Or, here's a menu of topics and quizzes from the MathGuide.com algebra site:

I could repeat this for many other cases, such as: the CUNY list of elementary algebra topics, tables of contents for most algebra books, etc., etc. It's weird and to my OCD brothers and sisters surely it's a bit distracting and frustrating.

There should be a name for this. The funny thing is that, to my current understanding, there's a perfectly serviceable name to make the distinction that we're reaching for here: "monic" means a polynomial with a lead coefficient of 1. So I've taken to, in my classes, referring to the initial or "basic" type (\(x^2 + bx + c\)) as a monic quadratic, and the more general or "advanced" type (\(ax^2 + bx + c\), \(a \ne 1\)) as a nonmonic quadratic. My students know they must learn proper names for everything, and so they pick this up as easily as anything else, and without complaint. Thereafter it's much easier to communally reference the different structures by their proper names.

Now: I must admit that I picked this up from Wikipedia and I've never, ever, seen it used in any mathematics textbook at any level. Perhaps someone could tell me if this is new, or nonstandard, or inaccurate. But even if that weren't the right term to distinguish a polynomial with lead coefficient 1, there should still be a name for this structure. We really should create a name, if necessary, and I'd be prone to make up my own name for something like that.

But "monic" fits perfectly and is delightfully short and descriptive. We should all start using "monic" more widely, and I'd love to start seeing it in major algebra textbooks.

Monday, October 10, 2016

Smaldino and McElreath write a paper which asserts that the problem of false-positive papers in science -- especially behavioral science -- is getting worse over time, and will continue to do so as long as we reward quantity of paper outputs:

To demonstrate the logical consequences of structural incentives, we
then present a dynamic model of scientific communities in which
competing laboratories investigate novel or previously published
hypotheses using culturally transmitted research methods. As in the real
world, successful labs produce more ‘progeny,’ such that their methods
are more often copied and their students are more likely to start labs
of their own. Selection for high output leads to poorer methods and
increasingly high false discovery rates. We additionally show that
replication slows but does not stop the process of methodological
deterioration. Improving the quality of research requires change at the
institutional level.

Quotes Campbell's Law: "The more any quantitative social indicator is used for social
decision-making, the more subject it will be to corruption pressures and
the more apt it will be to distort and corrupt the social processes it
is intended to monitor."